On the squashed seven-sphere operator spectrum

We derive major parts of the eigenvalue spectrum of the operators on the squashed seven-sphere that appear in the compactification of eleven-dimensional supergravity. These spectra determine the mass spectrum of the fields in $Ad{S}_4$ and are important for the corresponding $\mathcal{N}=1$ supermultiplet structure. This work is a continuation of the work in [1] where the complete spectrum of irreducible isometry representations of the fields in $Ad{S}_4$ was derived for this compactification. Some comments are also made concerning the $G_2$ holonomy and its implications on the structure of the operator equations on the squashed seven-sphere.